Water Hammer - Expression For The Excess Pressure Due To Water Hammer

Expression For The Excess Pressure Due To Water Hammer

When a valve with a volumetric flow rate Q is closed, an excess pressure δP is created upstream of the valve, whose value is given by the Joukowsky equation:

In this expression:

• overpressurization δP is expressed in Pa;
• Q is the volumetric flow in m3/s;
• Z_h is the hydraulic impedance, expressed in kg/m4/s.

The hydraulic impedance Z_h of the pipeline determines the magnitude of the water hammer pulse. It is itself defined by:

with:

• ρ the density of the liquid, expressed in kg/m3;
• A cross sectional area of the pipe, m2;
• B_eff effective modulus of compressibility of the liquid in the pipe, expressed in Pa.

The latter follows from a series of hydraulic concepts:

• compressibility of the liquid, defined by its adiabatic compressibility modulus B_l, resulting from the equation of state of the liquid generally available from thermodynamic tables;
• the elasticity of the walls of the pipe, which defines a modulus of equivalent compressibility B_eq. In the case of a pipe of circular cross section whose thickness e is small compared to the diameter D, the equivalent modulus of compressibility is given by the following formula: ; in which E is the Young's modulus (in Pa) of the material of the pipe;
• possibly compressibility B_g of gas dissolved in the liquid, defined by:
  • γ being the ratio of specific heats of the gas
  • α the rate of ventilation (the volume fraction of undissolved gas)
  • and P the pressure (in Pa).

Thus, the effective compressibility modulus is:

As a result, we see that we can reduce the water hammer by:

• increasing the pipe diameter at constant flow, which reduces the inertia of the liquid column;
• choosing to use a material with a reduced Young's modulus;
• introducing a device that increases the flexibility of the entire hydraulic system, such as a hydraulic accumulator;
• where possible, increasing the percentage of undissolved air in the liquid.